84] TWO SURFACES OF THE SECOND ORDER. 493 
and these values give (with the same signification as before of f g, h, l, m, n) 
2 (AC — B 2 ) = Aa 2 +Bb 2 + Cc 2 + 2Fbc +2Gca +2Hab +2Lad +2Mbd +2Ned +Dd\ 
2 (BD - C 2 ) = A a! 2 + Bb' 2 + Gc' 2 A 2Fb'c' + 2Gc'a' + 2Ha'V A 2La'd' A 2Mb‘d! + 2Nc'd’ A Dd' 2 , 
AD - BC = Aaa' + Bbb' A Ccc' + F(bc’ a b'c) + G (ca' A da) + H (ab' + a'b) 
+ L (ad' + a'd) + M (bd' + b'd) -f N (cd' + c'd) + Ddd', 
where A = n 2 y 4 + mV 4 + f 2 w 4 + 2fmz 2 w 2 — 2fny 2 w 2 A 2nmy 2 z 2 , 
B = l 2 z i + n 2 od + g 2 w 4 + 2gnx 2 w 2 — 2glz 2 w 2 + 2 In z 2 x 2 , 
C = m 2 x 4 -f l 2 y 4 + h 2 w 4 + 2hly 2 w 2 — 2hmx 2 w 2 + 2mlx 2 y 2 , 
D = f 2 x* + g 2 y 4 + JPz 4 — 2ghy 2 z 2 — 2hfz 2 x 2 — 2fg y 2 z 2 , 
F = l 2 y 2 z 2 , 
G = m 2 z 2 x 2 , 
H = n 2 x 2 y 2 , 
L = f 2 x 2 w 2 , 
M = g 2 y 2 w 2 , 
N = h 2 z 2 w 2 ; 
and then 4 (AC - B 3 ) (BD - C 2 ) - (AD - BC) 2 - 
(BC - F 2 ) f 2 + (CA - G 2 ) g 2 + (AB - H 2 ) h 2 +(AD- L 2 ) l 2 + (BD - M 2 ) m 2 4- (CD - N 2 ) n? 
+ 2 (GH-AF)gh +2 (HF-BG)lif f2 (FG-CH)fg 
- 2 (MN - DF) mn -2 (NL - DG ) nl - 2 (LM - DH) Im 
+ 2 (AM - LH) Ih +2 (BN — NF ) mf + 2 (CL - NG) ng 
- 2 (AN - LG ) Ig - 2 (BL - MH) mh-2 (CM - NF) nf 
+ 2 (NH - MG) If A 2 (LF - NH )mg + 2 (MG - LF ) nh, 
Substituting the values of A, B, &c., in this expression, the result after all reductions is 
f 2 m 2 n 2 x & -f- g 2 n 2 l 2 y s + li 2 l 2 m 2 z* -f f 2 g 2 h 2 w s 
-t 2gl 2 n (mg — nh) y e z 2 +2hm 2 l(nh — lf)z 6 x 2 + 2fn 2 m (If — mg) x 6 y 2 
— 2 hPm (mg — nh) y 2 z? — 2fm 2 n (nh — If) z 2 oA — 2gn 2 l (If — mg) x 2 y 6 
+ 2f 2 mn (mg — nh) xhif + 2 g 2 nl (nh — If) y 6 w 2 + 2 h 2 lm (If — mg) Aw 2 
— 2f 2 gh (mg - nh) xhif — 2fg 2 h (nh — If) y 2 w 6 — 2fgh 2 (If — mg) z 2 w 6 
+ /' (Pf* ~ Qghmn) mjV + g 2 (m 2 g 2 — Ghfnl) w 4 f + h 2 (n 2 h 2 — Glmfg) w*z 4 
+ P (Pf 2 — 6ghmn) ifz 4 + m 2 (m 2 g 2 — 6hfnl) z 4 F + n 2 (n 2 h 2 — Qlmfg) cAy 4 
+ 2gh (ghmn — 3f 2 P) vf y 2 z 2 + 2lif(hfnl — 3g 2 m 2 ) vAz 2 x 2 +2fg (fglm — 3h 2 n 2 ) w*x 2 y 2 
A 2hm (ghmn — 3f 2 P) z 4 x 2 w 2 A 2fn (Ifni — Sg 2 m 2 ) oAy 2 w 2 + 2gl (fglm — 3lPn 2 ) y 4 z 2 w 2 
— 2gn (ghmn — 3f 2 l 2 ) y 4 x 2 w 2 — 2hi (hfnl — 3g 2 m 2 ) z i y 2 w 2 — 2fm (fglm — 3h 2 iP) oAz 2 w 2 
— 2nm (ghmn — 3f 2 P) oAy 2 z 2 — 2nl (hfnl — 3g 2 m 2 ) y 4 z 2 x 2 — 2 Im (fglm — 3h 2 ri 2 ) z 4 x 2 y' 1 
— 2 (mg — nh) (nh — If ) (If — mg) x 2 y 2 z 2 w 2 = 0,